It has been known in the art from U.S. Pat. No. 3,646,155 etc. to produce a cross-linked polyethylene by reacting a polyethylene with a free radical generating compound and an organic silane having the general formula RR'SiY.sub.2 (wherein R represents a monovalent olefinically unsaturated hydrocarbon or hydrocarbonoxy radical, each Y represents a hydrolyzable organic radical and R' is a monovalent hydrocarbon radical except aliphatically unsaturated hydrocarbon radicals, or is a Y radical) to form a silane-grafted polyethylene, and subjecting said silane-grafted polyethylene to action of warm water or steam in the presence of a silanol condensation catalyst to obtain a crosslinked polyethylene. According to the prior technical concept, no polyethylene having a greater degree of crosslinking would be produced by merely heating a silane-grafted polyethylene in the presence of a silanol condensation catalyst. Thus, it has been common in the prior art to adopt a process that a silane-grafted polyethylene is mixed with a silanol condensation catalyst followed by being molded with heating, and the resulting shape is soaked in a warm aqueous bath or a steam chamber to crosslink. Otherwise, there has been practiced a process wherein a sinane-grafted polyethylene is added with a silanol condensation catalyst and water, the resulting mixture is fed to an extruder, and the crosslinking takes place simultaneously with shaping by extrusion. However, the former process has the drawback that a warm aqueous bath or a steam chamber is required to install and, additionally, the step of contacting the extruded article with warm water or steam must be manually operated to increase the cost of production. The latter process also has drawbacks in that the moisture fed to the extruder forms bubbles of steam which emerge on the surface of the extrudate and a poor extruded article having a high surface roughness is obtained. In some cases where the resin is crosslinked to a certain extent by the heat generated in course of shaping in the extruder, the resin unfavorably cannot melt to flow and the extrudate cannot be obtained.